Mental Mathematics – Large Roots

We’re here today to talk about Banyan Grove tree root systems… NOT.

 

So your friend comes to you and asks you, “what’s the 6th root of 244,140,625?”  Most of us are going to respond “lol wut?”, or “how the hell am I supposed to know?!”, or maybe “let me get my calculator.”  But there are a few people who can look at that number and say “oh, that’s 25. easy.”  How do they do it?

Taking large roots of large numbers may seem darn near impossible when you first consider it, but I’m going to teach you here today how to do it. You don’t need a high IQ or an amazing memory to do this, just a little bit of practice and a tiny bit of memorization. So you Average Joes, stun your friends and get girls with the fantastic power of mental mathematics today!

 

Yes.  Frightening.

So, to begin, we’re going to look at the last digit of the large number we have to take a root of. This digit could be zero, one, two, three, four, five, etc. up to nine. However, some digits are easier to work with than others. Let’s multiply some single digit numbers together and see what happens.

 

1 = 1
1 * 1 = 1
1 * 1 * 1 = 1
1^n = 1

One is always one, no matter how many times you multiply it by itself. 1, 1, 1, … etc.

2 = 2
2 * 2 = 4
2 * 2 * 2 = 8
2 * 2 * 2 * 2 = 16
2 * 2 * 2 * 2 * 2 = 32

Dealing only with last digit, we can treat this as a repeating sequence. 2, 4, 8, 6, 2, … etc.

3 = 3
3 * 3 = 9
3 * 3 * 3 = 27
3 * 3 * 3 * 3 = 81
3 * 3 * 3 * 3 * 3 = 243

Dealing only with the last digit we can treat this as a repeating sequence. 3, 9, 7, 1, 3, … etc.

4 = 4
4 * 4 = 16
4 * 4 * 4 = 64

Dealing only with the last digit we can treat this as a repeating sequence. 4, 6, 4, … etc.

5 = 5
5 * 5 = 25
5 * 5 * 5 = 125

Dealing only with the last digit we can treat this as a repeating sequence. 5, 5, 5, … etc.

6 = 6
6 * 6 = 36
6 * 6 * 6 = 216

Dealing only with the last digit we can treat this as a repeating sequence. 6, 6, 6, 6 … etc.

7 = 7
7 * 7 = 49
7 * 7 * 7 = 273
7 * 7 * 7 * 7 = 1911
7 * 7 * 7 * 7 * 7 = 16807

Dealing only with the last digit we can treat this as a repeating sequence. 7, 9, 3, 1, 7, … etc.

8 = 8
8 * 8 = 64
8 * 8 * 8 = 512
8 * 8 * 8 * 8 = 4096
8 * 8 * 8 * 8 * 8 = 32768

Dealing only with the last digit we can treat this as a repeating sequence. 8, 4, 2, 6, 8, … etc.

9 = 9
9 * 9 = 81
9 * 9 * 9 = 729

Dealing only with the last digit we can treat this as a repeating sequence. 9, 1, 9, … etc.

10 = 10
10 * 10 = 100
10 * 10 * 10 = 1000

Dealing only with the last digit we can treat this as a repeating sequence. 0, 0, 0, … etc.

 

So let’s arrange these sequences we found for what the last digit will be side by side.
1: 1, 1, 1, … etc.
2: 2, 4, 8, 6, 2, … etc.
3: 3, 9, 7, 1, 3, … etc.
4: 4, 6, 4, 6, … etc.
5: 5, 5, 5, … etc.
6: 6, 6, 6, … etc.
7: 7, 9, 3, 1, 7, … etc.
8: 8, 4, 2, 6, 8, … etc.
9: 9, 1, 9, 1, … etc.
10: 0, 0, 0, 0, … etc.

Based on this information we can make a conclusion about what the last digit of a root of a number is based on the number and the number of roots we’re taking.

 

 

Examples:

Find the fifth root of 28,629,151:

-The root can’t end in three, because position 5 in the sequence for the number 3 is three.
3, 9, 7, 1, 3
-The root can’t end in seven. 7, 9, 3, 1, 7
-The root can’t end in nine. 9, 1, 9, 1, 9
-The root must end in a one. 1, 1, 1, 1, 1

From here we can use another technique to get a better idea of what the root is. It’s a type of order of magnitude estimate.

28,629,151 can be presented in scientific notation as 2.8629151 * 10^7. We’re taking the fifth root of this number, and we can say that this number is fairly close to 10^7. So by magnitude estimations, the root should have 7/5 zeroes.† This is 1.4 zeroes. This may seem really weird to you. How can you have 1.4 zeroes?! Think about it this way, a number with two zeroes is more than 99 and less than 1000. A number with one zero is more than 9 and less than 100. So a number with 1.4 zeroes is more than 9 and less than 99.

So right now we have brought things down to a fairly small range of numbers. The answer could be 11, 21, 31, 41, 51, 61, 71, 81, or 91. But how do bring things down more? Let’s look again at the number 1.4. This number is closer to 1 than it is to 2, but it’s also pretty darn close to being right in between the two (1.5). So we can rule out the really large numbers and the really small ones. Let’s rule out 11, 51, 61, 71, 81, and 91. That leaves 21, 31, or 41 as our answer. Would you look at that. We haven’t done any real division or massive multiplication, but we’ve gotten the answer down to three possible numbers. Let’s go with the middle one and make a quick calculation.

31^5 ~ 30^5 = 3 ^ 5 * 10^5 = 9 * 9 * 3 * 100,000 = 243 * 100,000 = 24,300,000.

That’s close enough, the answer must be 31! (and if you get out your calculator, you’ll find out it is)

 

†Note: this method of magnitude estimation becomes less accurate as you start to deal with larger numbers and larger roots.

 

Find the third root of 373,248:
3rd number in sequence is 8: root must end in 2.
3.73248 * 10^5
5/3 = 1.66
1.5 > 1.66 > 2.0
52, 62, or 72
62: 6 ^ 3 * 10^3 = 216 * 1000 = 216,000
72: 7 ^ 3 * 10^3 = 343 * 1000 = 343,000
72.

 

Find the fifth root of 2,373,046,875:
5th number in sequence is 5: root must end in 5.
2.373046875 * 10^9
9/5 = 1.8
9 < r < 99
55, 65, 75, 85
85 ^ 5 ~ 8^5 * 10^5 = 64 * 64 * 8 * 10^5 = 512 * 64 * 10^5 = 4096 * 8 * 10^5 = 8092 * 4 * 10^5 = 16,186 * 2 * 10^5 = 32,372 * 10^5 = 3,237,200,000
Too big by just a bit.
75.

 

Find the fifth root of 656,356,768:
5th number in sequence is 8: root must end in 8.
6.56356768 * 10^8
8/5 = 1.6
9 < r < 99
48, 58, 68, 78
48 ^ 5 ~ 50 ^ 5 = 5 ^ 5 * 10^5 = 25 * 25 * 5 * 10^5 = 3125 * 10^5 = 312,500,000
Too small by just a bit.
58.

 

Find the sixth root of 46,656,000,000:
6th number in sequence is 0: root must end in 0.
4.6656 * 10^10
10/6 = 1.66
9 < r < 99
40, 50, 60, 70
46,656 ends in 6, sixth root ends in 6. Could be 40 or 60.
4 ^ 6 = 16 * 16 * 16 = 256 * 16 = 4096
Doesn’t match.
60.

 

Find the ninth root of 427,929,800,129,788,411:
9th number in sequence is 1: root must end in 1.
~4.279 * 10^17
17/9 = 1 and 8/9 ~ 1.88
9 < r < 99
71, 81, 91
91.

 

Find the fourth root of 38,416:
4th number in sequence is 6: root could end in 2, 4, 6, or 8.
3.8416 * 10^4
4/4 = 1.00
Because we got 1.00, we know it has to be very close to 10.
12, 14, 16
14.

 

Find the fifth root of 147,008,443:
5th number in sequence is 3: root must end in 3.
1.47 * 10^8
8/5 = 1.6
9 < r < 99
43, 53, 63, 73
53^5 ~ 50^5 = 5^5 * 10^5 = 3125 * 10^5 = 312,500,000
A little too big.
43.

 

Find the seventh root of 55,784,660,123,648:
7th number in sequence is 8: root must end in 2.
5.5784 * 10^13
13/7 = 1 and 6/7 ~ 1.85
72, 82, 92
92.

 

Find the third root of 60,236,288:
3rd number in sequence is 8: root must end in 2.
6.0236 * 10^7
7/3 = 2.33
330 > r > 400
332, 342, 352, 362, 372, 382, 392
Dealing with larger roots (with 2 or more zeroes) becomes more complicated, and will require additional methods which I may devise and present later on.
392.

 

As you can see, this method works. If you learn this method and practice a bit you should be able to get up to the 10th root of any number up to 10^20 in under two minutes, assuming the root is less than 3 digits.

 

Have fun you little math wizards.